What if wheels were square?

Bikes, cars and buses all have round wheels.

What if wheels were square? Could you still ride?

Look at any bike, car, train or shopping cart and you will find round wheels. Why round, and not some other shape?

In this lab you can swap a round wheel for a square one and roll it along a road. Will it work at all? And is there any kind of road that would make a square wheel roll smoothly?

A wheel's job is to carry its axle forward at a steady height while rolling without slipping. A circle does this on flat ground because every point of its rim is the same distance from the center.

This lab asks two questions. How badly does a regular polygon, here a square, perform on a flat road? And what road shape, if any, would keep a square wheel's axle perfectly level?

A round wheel's middle stays at the same height.

So the ride is smooth. No bumps!

Every point on the edge of a circle is the same distance from its middle. So as a round wheel rolls, its axle, the rod through its center, stays at exactly the same height above a flat road.

That is why a round wheel gives a smooth ride. In the model, the circle's axle bounce is zero and it loses no energy as it rolls.

For a circle of radius R on a flat road, the contact point is always directly below the center at distance R, so the axle height is constant and rolling is smooth.

In the model a circle has zero axle bounce and zero impact loss. Polygons approach this as their number of sides grows: the model gives a bounce of 4.0 cm for a hexagon, 2.3 cm for an octagon and 1.0 cm for a 12-sided wheel, all with R = 0.3 m.

Olly puts on a square wheel. Bump, bump, bump!

Then Olly rolls it on a special wavy road. Now it rolls smoothly!

Choose the square wheel on a flat road. It rocks up onto each corner and drops down again, lifting its axle almost 9 centimeters every time, a little more than twice in each meter. Each time a corner hits the road, the model says it loses about 94 percent of its motion energy.

Now switch on the matching bumpy road. The road is made of curved humps that fit the square's sides. The axle stops bouncing, and the square rolls smoothly with no energy lost at the corners.

On a flat road the square pivots over each corner, so its axle height is R cos φ for |φ| ≤ 45°, rising and falling by 8.8 cm with 2.36 impacts per meter. Conserving angular momentum about each new corner gives ω₊/ω₋ = 0.25, so 93.75 percent of the kinetic energy is lost per impact.

On the matching road, inverted catenaries y = −a cosh(x/a) with a = R cos 45°, the axle stays level and the corners meet the road in the troughs between humps. The model gives 2.67 humps per meter and no impact loss.

A square's corners stick out farther than its sides.

So on flat ground, the middle goes up and down. The wavy road fills in the gaps.

A square's corners are farther from its middle than the middle of its sides are. On a flat road, the wheel tips up over a corner, lifting the axle, then flops down onto the next side.

The matching road works because its humps are exactly the right shape. A hump rises under the middle of each flat side and dips where the corners land, so the distance from the axle to the road never changes. That special curve is an upside-down catenary, the shape a chain makes when it hangs between two points.

For a level axle, the contact point must stay directly beneath the center, and rolling without slipping requires the road's arc length to match the distance the contact point travels along the side. Solving these conditions gives the inverted catenary; each arc spans 2a·asinh(tan(π/n)) and turns the wheel through 2π/n.

The model's limits: a rigid wheel that never slips or bounces, an axle pushed along at a set speed, and a simple impact rule. Its loss figure is an idealization of a perfectly stiff wheel. The model offers no matching road for a triangle: its corners would dig into the next hump.

There is a real bike with square wheels! It rides on a special bumpy road.

Anyone can try it at a college in Minnesota.

This is not just a math trick. In 1997 a mathematics professor, Stan Wagon, and a bike mechanic, Loren Kellen, built a square-wheeled bike at Macalester College in Minnesota that rolls smoothly over a road of upside-down catenaries.

The first one wore out and was rebuilt in 2004 with stronger parts. The college says the new ride is so smooth you can't tell you aren't on round wheels. Visitors are allowed to ride it, but only on its own special road.

Macalester College reports that Stan Wagon and Loren Kellen built a square-wheel vehicle in 1997 that rolls over a road of inverted catenaries, replaced in April 2004 with more robust materials; its odometer showed just over 15 miles between September 2004 and September 2005.

Hall and Wagon ("Roads and Wheels", Mathematics Magazine, 1992) discuss the differential equation that matches a road to other wheel shapes (Harvey Mudd Math Fun Facts), and Macalester credits G. B. Robison (1960) with first realizing that a square could roll on catenaries. Round wheels win in daily life because a flat road suits them; a square wheel needs a road made just for it.